14 C.Hirlimann feedback,allowing the device to run as an oscillator.In order for the oscillation to operate in a single mode the size of the box had to be of the order of a few wavelengths,i.e.a few centimeters.A large number of research groups tried to transpose the technique to the visible range using appropriate gain media. They were stopped by the necessity to design an optical box having a volume of the order of A3,which in the visible means 0.1 um3.A tractable much larger box would have had a large number of modes fitting the gain bandwidth,and this was expected to create mode beating which in turn would destroy the build-up of a coherent oscillation. In the following years,Schawlow and Townes [1.2],as well as Basov and Prokhorov [1.3],came up with calculations showing that the number of modes in an optical cavity could be greatly reduced by confining light in only one dimension to create a feedback.The Fabry-Perot resonator then came into the picture. A gain medium would be put between the two high-reflectivity 100 % mirrors of a Fabry-Perot interferometer so that a coherent wave could be constructed after several round trips of the light through the amplifier.At the starting time of the device,when the inverted population was established in the gain medium,a unique spontaneously emitted photon propagating along the cavity axis would start stimulated emission,increasing the number of co- herent photons.If,after a round trip between the mirrors,the gain was larger than the losses then the intensity of the visible electromagnetic wave would increase as an exponential function after each round trip,and a self-sustained oscillation would start.But in a cavity,owing for example to diffraction at the edge of the mirrors or spurious reflections and absorption,photons are lost. The value of the gain which overcomes the losses is called the laser threshold. 1.6.2 Geometric Point of View We will now focus on the properties of an optical cavity,and specifically look for the necessary conditions that must be fulfilled so that the cavity can accommodate an infinite number of round trips of the light.The cavity under consideration consists simply of two concave,spherical mirrors with radii of curvature Ri and R2,spaced by a length L(Fig.1.12).From a geometric point of view,Fig.1.12 shows that the light ray coincident with the mirrors'axis will repeat itself after an arbitrary number of back-and-forth reflections from the mirrors.Other rays may or may not escape the volume defined by the two mirrors.A cavity is said to be stable when there exists at least one family of rays which never escape.When there is no such ray the cavity is said to be unstable;any ray will eventually escape the volume defined by the mirrors. A very simple geometric method allows one to predict whether a given cavity is stable or not 1.4.Consider now the cavity defined in Fig.1.13.Two circles having their centers on the axis of the cavity are drawn with their diameters equal to the radii of curvature of the mirrors so as to be tangent to the reflecting face of the mirrors.The center of each is coincident with the
14 C. Hirlimann feedback, allowing the device to run as an oscillator. In order for the oscillation to operate in a single mode the size of the box had to be of the order of a few wavelengths, i.e. a few centimeters. A large number of research groups tried to transpose the technique to the visible range using appropriate gain media. They were stopped by the necessity to design an optical box having a volume of the order of λ3, which in the visible means 0.1 μm3. A tractable much larger box would have had a large number of modes fitting the gain bandwidth, and this was expected to create mode beating which in turn would destroy the build-up of a coherent oscillation. In the following years, Schawlow and Townes [1.2], as well as Basov and Prokhorov [1.3], came up with calculations showing that the number of modes in an optical cavity could be greatly reduced by confining light in only one dimension to create a feedback. The Fabry–P´erot resonator then came into the picture. A gain medium would be put between the two high-reflectivity (≈ 100 %) mirrors of a Fabry–P´erot interferometer so that a coherent wave could be constructed after several round trips of the light through the amplifier. At the starting time of the device, when the inverted population was established in the gain medium, a unique spontaneously emitted photon propagating along the cavity axis would start stimulated emission, increasing the number of coherent photons. If, after a round trip between the mirrors, the gain was larger than the losses then the intensity of the visible electromagnetic wave would increase as an exponential function after each round trip, and a self-sustained oscillation would start. But in a cavity, owing for example to diffraction at the edge of the mirrors or spurious reflections and absorption, photons are lost. The value of the gain which overcomes the losses is called the laser threshold. 1.6.2 Geometric Point of View We will now focus on the properties of an optical cavity, and specifically look for the necessary conditions that must be fulfilled so that the cavity can accommodate an infinite number of round trips of the light. The cavity under consideration consists simply of two concave, spherical mirrors with radii of curvature R1 and R2, spaced by a length L (Fig. 1.12). From a geometric point of view, Fig. 1.12 shows that the light ray coincident with the mirrors’ axis will repeat itself after an arbitrary number of back-and-forth reflections from the mirrors. Other rays may or may not escape the volume defined by the two mirrors. A cavity is said to be stable when there exists at least one family of rays which never escape. When there is no such ray the cavity is said to be unstable; any ray will eventually escape the volume defined by the mirrors. A very simple geometric method allows one to predict whether a given cavity is stable or not [1.4]. Consider now the cavity defined in Fig. 1.13. Two circles having their centers on the axis of the cavity are drawn with their diameters equal to the radii of curvature of the mirrors so as to be tangent to the reflecting face of the mirrors. The center of each is coincident with the
1 Laser Basics 15 R2 R1 M M2 十 Fig.1.12.Optical cavity consisting of two concave mirrors.Two ray paths are shown.The axial ray is stable,the other is not R2 R1 2 F2 M1 2-0 M2 Fig.1.13.Concave-convex laser cavity.Geometric determination of the stability of the cavity 1.4 real or virtual focus point of the mirror.The straight line joining the points of intersection of the two circles crosses the axis of the cavity at a point which defines the position of the beam waist of the first-order transverse mode.If the two circles do not cross,the cavity is unstable. 1.6.3 Diffractive-Optics Point of View Geometrical optics is not enough to estimate quantitatively the properties of the modes which may be established in a Fabry-Perot cavity.The full
1 Laser Basics 15 R1 R2 M1 M2 L Fig. 1.12. Optical cavity consisting of two concave mirrors. Two ray paths are shown. The axial ray is stable, the other is not Fig. 1.13. Concave–convex laser cavity. Geometric determination of the stability of the cavity [1.4] real or virtual focus point of the mirror. The straight line joining the points of intersection of the two circles crosses the axis of the cavity at a point which defines the position of the beam waist of the first-order transverse mode. If the two circles do not cross, the cavity is unstable. 1.6.3 Diffractive-Optics Point of View Geometrical optics is not enough to estimate quantitatively the properties of the modes which may be established in a Fabry–P´erot cavity. The full
6 C.Hirlimann calculation of these modes is rather tedious and we will concentrate only on some of the most important features.In a Fabry-Perot interferometer light bounces back and forth from one mirror to the other with a constant time of flight,so that its dynamics is periodic by nature.A wave propagating inside a cavity remains unchanged after one period in the simple case in which the polarization direction does not change.The propagation is governed by diffraction laws because of the finite diameter of the mirrors,but also because of the presence of apertures inside the cavity;the most common aperture is simply the finite diameter of the gain volume.From a mathematical point of view,the radial distribution of the electric field in a given mode inside the cavity is described through a two-dimensional spatial Fourier transformation. For the periodicity condition to be respected it is therefore necessary that the function describing the radial distribution is its own Fourier transform.The Gaussian function is its own Fourier transform and is therefore the very basis of the transverse electromagnetic structure for spherical-mirror cavities. It can be shown that the electric field distribution of the fundamental transverse mode in a spherical-mirror cavity can be written as ={-(e+)-} Wo x2+y2 ×exp-W2(a)' (1.16) where (1.17） Here R(z)is the radius of curvature of the wave surface (Fig.1.14),(z)is the phase as a function of the distance z, w)-w(+()月） (1.18) is the radius of the beam and k= 入 (1.19) is the propagation factor in vacuum. The propagation origin z=0 is chosen to be coincident with the position of the minimum radius of the beam,the beam waist Wo.When z=0 then R(0)=(0)=oo,the wave surface is plane and the electric field amplitude E(x,y,0)=Eo exp-(22+y2/Wo) (1.20) decays as a Gaussian function along the radius of the beam
16 C. Hirlimann calculation of these modes is rather tedious and we will concentrate only on some of the most important features. In a Fabry–P´erot interferometer light bounces back and forth from one mirror to the other with a constant time of flight, so that its dynamics is periodic by nature. A wave propagating inside a cavity remains unchanged after one period in the simple case in which the polarization direction does not change. The propagation is governed by diffraction laws because of the finite diameter of the mirrors, but also because of the presence of apertures inside the cavity; the most common aperture is simply the finite diameter of the gain volume. From a mathematical point of view, the radial distribution of the electric field in a given mode inside the cavity is described through a two-dimensional spatial Fourier transformation. For the periodicity condition to be respected it is therefore necessary that the function describing the radial distribution is its own Fourier transform. The Gaussian function is its own Fourier transform and is therefore the very basis of the transverse electromagnetic structure for spherical-mirror cavities. It can be shown that the electric field distribution of the fundamental transverse mode in a spherical-mirror cavity can be written as E(x, y, z) = E0 W0 W(z) exp − i k z + x2 + y2 2R(z) − Φ(z) × exp −x2 + y2 W2(z) , (1.16) where R(z) = z 1 + πW2 0 λz 2 , Φ(z) = tan−1 λz πW2 0 . (1.17) Here R(z) is the radius of curvature of the wave surface (Fig. 1.14), Φ(z) is the phase as a function of the distance z, W2(z) = W2 0 1 + λz πW2 0 2 (1.18) is the radius of the beam and k = 2π λ (1.19) is the propagation factor in vacuum. The propagation origin z = 0 is chosen to be coincident with the position of the minimum radius of the beam, the beam waist W0. When z = 0 then R(0) = Φ(0) = ∞, the wave surface is plane and the electric field amplitude E(x, y, 0) = E0 exp − x2 + y2/W2 0 (1.20) decays as a Gaussian function along the radius of the beam.
1 Laser Basics 17 Wo w(z) R(z) 2=0 Fig.1.14.Schematic structure of a Gaussian light beam in the vicinity of a focal volume.The y axis is perpendicular to the paper sheet.The z=0 origin is at the minimum radius Wo of the beam Along the axis,the radius of curvature of the wave surface R(z)varies as a hyperbola and its asymptotes make an angle 0 with the axis such that tan=A/rWo.This angle is a good definition of the beam divergence. For large z the hyperbola may be replaced by its asymptotes and the radius of curvature varies linearly with the distance z;R(z)z when z goes to infinity in (1.17).In this long-distance regime the amplitude of the electric field varies as the inverse of the beam radius,i.e.E(z)W-1(z),along z,and as a Gaussian function along the radius.Putting aside the Gaussian attenuation,the fundamental mode behaves like a spherical wave.Such a wavefront has the right shape to fit nicely the surface of spherical mirrors. This structure of the fundamental transverse mode is referred to as TEMoo (fundamental transverse electric and magnetic mode).Many other transverse modes can propagate inside the cavity;they can be expressed as the super- position of higher-order fundamental modes TEMnm.These modes can be calculated by multiplying the fundamental lowest-order mode (1.16)by the Hermite polynomials of integer orders n and m, ()() (1.21) and multiplying the phase term (z)by (1+n+m). 1.6.4 Stability of a Two-Mirror Cavity The problem is now to find which Gaussian mode with a far-field spherical behavior can fit a given pair of spherical mirrors with radii Ri and R2,spaced by a distance L.In Fig.1.15,the mirror positions z1 and z2 are measured from the yet unknown position of the beam waist.For a cavity to be stable it must be able to accommodate a mode in which spherical wavefronts will fit the reflecting surfaces of the two spherical mirrors.From a formal point of view,one simply has to make the radii of curvature of the wavefront,given by (1.17),equal to the radii of curvature of the mirrors;adding the conservation of length leads to the three equations
1 Laser Basics 17 Fig. 1.14. Schematic structure of a Gaussian light beam in the vicinity of a focal volume. The y axis is perpendicular to the paper sheet. The z = 0 origin is at the minimum radius W0 of the beam Along the axis, the radius of curvature of the wave surface R(z) varies as a hyperbola and its asymptotes make an angle θ with the axis such that tan θ = λ/πW0. This angle is a good definition of the beam divergence. For large z the hyperbola may be replaced by its asymptotes and the radius of curvature varies linearly with the distance z; R(z) ≈ z when z goes to infinity in (1.17). In this long-distance regime the amplitude of the electric field varies as the inverse of the beam radius, i.e. E(z) ≈ W−1(z), along z, and as a Gaussian function along the radius. Putting aside the Gaussian attenuation, the fundamental mode behaves like a spherical wave. Such a wavefront has the right shape to fit nicely the surface of spherical mirrors. This structure of the fundamental transverse mode is referred to as TEM00 (fundamental transverse electric and magnetic mode). Many other transverse modes can propagate inside the cavity; they can be expressed as the superposition of higher-order fundamental modes TEMnm. These modes can be calculated by multiplying the fundamental lowest-order mode (1.16) by the Hermite polynomials of integer orders n and m, Hn √2 x W(z) , Hm √2 y W(z) , (1.21) and multiplying the phase term Φ(z) by (1 + n + m). 1.6.4 Stability of a Two-Mirror Cavity The problem is now to find which Gaussian mode with a far-field spherical behavior can fit a given pair of spherical mirrors with radii R1 and R2, spaced by a distance L. In Fig. 1.15, the mirror positions z1 and z2 are measured from the yet unknown position of the beam waist. For a cavity to be stable it must be able to accommodate a mode in which spherical wavefronts will fit the reflecting surfaces of the two spherical mirrors. From a formal point of view, one simply has to make the radii of curvature of the wavefront, given by (1.17), equal to the radii of curvature of the mirrors; adding the conservation of length leads to the three equations
18 C.Hirlimann R2 R1 M Z=0 十 2十 Fig.1.15.Schematic diagram of a simple transverse Gaussian beam fitting a two- spherical-mirror cavity =-- (1.22) 21 R2=+22+ 强 (1.23) 22 L=22-21, (1.24) where zR =TWo/A,called the Rayleigh range,is the distance,measured from the beam waist position,where the radius of the beam is equal to v2 Wo.This length defines the focal volume,which,to first order in z,is almost cylindrical. These simple equations have been generally solved using the special cavity parameters L 9g1=1-R L and g2 =1- R2' (1.25) tying the distances z1,22,zR to the geometric cavity parameters R1,R2,L. Solving the equations in this new notation leads to the following results: -the beam waist position measured from the mirror position 21= 92(1-9m)L, 22= 9n(1-92）-L: (1.26) 91+92-2g192 91+92-2g192 the beam waist radius, W哈= LX 9192(1-9192） (91+92-2g192)2 (1.27) the beam radius at the surface of the mirrors, w2= L入 92 元V91(1-g1g2) W吃= Lλ 91 TVg2(1-9192) (1.28)
18 C. Hirlimann R1 R2 M1 M2 z1 L z 2 z=0 Fig. 1.15. Schematic diagram of a simple transverse Gaussian beam fitting a twospherical-mirror cavity R1 = −z1 − z2 R z1 , (1.22) R2 = +z2 + z2 R z2 , (1.23) L = z2 − z1, (1.24) where zR = πW0/λ, called the Rayleigh range, is the distance, measured from the beam waist position, where the radius of the beam is equal to √2 W0. This length defines the focal volume, which, to first order in z, is almost cylindrical. These simple equations have been generally solved using the special cavity parameters g1 = 1 − L R1 and g2 = 1 − L R2 , (1.25) tying the distances z1, z2, zR to the geometric cavity parameters R1, R2, L. Solving the equations in this new notation leads to the following results: – the beam waist position measured from the mirror position z1 = g2(1 − g1) g1 + g2 − 2g1g2 L, z2 = g1(1 − g2) g1 + g2 − 2g1g2 L; (1.26) – the beam waist radius, W2 0 = Lλ π g1g2(1 − g1g2) (g1 + g2 − 2g1g2)2 ; (1.27) – the beam radius at the surface of the mirrors, W2 1 = Lλ π g2 g1(1 − g1g2) , W2 2 = Lλ π g1 g2(1 − g1g2) . (1.28)